Lesson 21 Expressions and equations

Episode 1

Reasoning Resources: None
Exploring all the different expressions which can be constructed using just two small numbers. Depending on their prior learning, this episode may take between 10 minutes in a whole class mode or up to 45 minutes in ‘mini-episodes’ Alternatively you could cover the agenda in two parts as suggested here.
Pair and group work on four operations and communativity
Using the vocabulary of expressions and operations.

Understanding that the equals sign has a meaning other than ‘the answer is’— this can be a stumbling block for some pupils.
Ask pairs of pupils each to choose a different number between 2 and 9.

How many calculations can you write, using each of the numbers once only and one of the four operations?

They write all the expressions they can.
Whole class sharing/discussion
Understanding of commutativity through practical demonstration.

Seeing the breaking up of a number as having many different possibilities.
Write the pupils’ calculations on the board and refer to them as ‘expressions’ explaining that the everyday word ‘sum’ is confusing because technically a sum only refers to addition. Pair up the expressions which use the same operation and discuss whether they are equal or not. Write = or # between them.

Focus on the equals sign and explain that here it is not giving the answer, but indicating that the left side is equal to the right side. For which operations does the order not matter?

Get pupils to explain this in their own words and tell them that the mathematical word for this is ‘commutativity: Relate it to ‘commuting’ and ‘commuters’ - going there and back both ways.
Pair and group work on breaking up and recombining numbers
Relating breaking up to partitioning when doing mental calculations — addition and multiplication. Write ‘4 + 5 ='on the board. Invite them to find expressions which equal this, giving an example 4 + 4 + 1. How many different expressions can you find?

They find different equivalent expressions. How does this relate to partitioning of numbers when doing mental calculations? e.g. 23+38=20+3+430+8=50+11=61.
Whole class sharing and discussion
Being reminded of the order of operations rules, particularly in relation to multiplication and addition.

Reinforcing the use of the equals sign.
Allow pupils to put into words the advantage of working separately with units and tens, whether on the page or mentally. Some pupils will prefer to break up only one number and complement the other numbers e.g. for 65 + 18 they may write 63+2+ 18. Discuss multiplication: 6 X 14 = 6x10+6 x4.

Emphasise that we have to do the second TIERS before adding the two terms. We may imagine brackets around the multiplication terms. Say that in mathematics we need a standard way of writing calculations that will be understood by everyone, without confusion. This is why we always multiply before adding (remind pupils about the order of operations which they will have learned earlier). Another convention we use is to write an equals sign when the whole of one side is equal to the whole of the other side, not just the first number.
Whole class reflection
Consolidating the main ideas of the lesson. Rehearse the advantages of looking at expressions in this way.


Thinking Mathematics Lessons Copyright © by Michael Shayer and Mundher Adhami. All Rights Reserved.

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